You sure? Probability of all corners being oriented is 1/27, correct permutation is 1/6.
1/(6*27) = 1/162.
E: Where you've gone wrong is CP. You've included parity (which is irrelevant when only looking at corners), and didn't remember that any AUF counts as corners being permuted. It shouldn't be 4!/2, it should be 4!/4.